Full-information estimation of heterogeneous agent models using macro and micro data

Liu, Laura; Plagborg-Møller, Mikkel Article Full-information estimation of heterogeneous agent models using macro and micro data Quantitative Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Liu, Laura; Plagborg-Møller, Mikkel (2023) : Full-information estimation of heterogeneous agent models using macro and micro data, Quantitative Economics, ISSN 1759-7331, The Econometric Society, New Haven, CT, Vol. 14, Iss. 1, pp. 1-35, https://doi.org/10.3982/QE1810 This Version is available at: https://hdl.handle.net/10419/296324 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/ Quantitative Economics 14 (2023), 1–35 1759-7331/20230001 Full-information estimation of heterogeneous agent models using macro and micro data Laura Liu Department of Economics, Indiana University Mikkel Plagborg-Møller Department of Economics, Princeton University We develop a generally applicable full-information inference method for heterogeneous agent models, combining aggregate time series data and repeated crosssections of micro data. To handle unobserved aggregate state variables that affect cross-sectional distributions, we compute a numerically unbiased estimate of the model-implied likelihood function. Employing the likelihood estimate in a Markov Chain Monte Carlo algorithm, we obtain fully efficient and valid Bayesian inference. Evaluation of the micro part of the likelihood lends itself naturally to parallel computing. Numerical illustrations in models with heterogeneous households or firms demonstrate that the proposed full-information method substantially sharpens inference relative to using only macro data, and for some parameters micro data is essential for identification. Keywords. Bayesian inference, data combination, heterogeneous agent models. JEL classification. C11, C32, E1. 1. Introduction Macroeconomic models with heterogeneous agents have exploded in popularity in recent years.1New micro data sets—including firm and household surveys, social security and tax records, and censuses—have exposed the empirical failures of traditional representative agent approaches. The new models not only improve the fit to the data, but also make it possible to meaningfully investigate the causes and consequences of inequality among households or firms along several dimensions, including endowments, financial constraints, age, size, location, etc. Laura Liu: [email protected] Mikkel Plagborg-Møller: [email protected] We are grateful for helpful comments from two anonymous referees, Adrien Auclert, Yoosoon Chang, Marco Del Negro, Simon Mongey, Hyungsik Roger Moon, Ulrich Müller, Jonathan Payne, Frank Schorfheide, Neil Shephard, Thomas Winberry, and participants at various seminars and conferences. Plagborg-Møller acknowledges that this material is based upon work supported by the NSF under Grant 1851665. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the NSF. 1For references and discussion, see Krueger, Mitman, and Perri (2016), Ahn, Kaplan, Moll, Winberry, and Wolf (2017), and Kaplan and Violante (2018). ©2023 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at http://qeconomics.org.https://doi.org/10.3982/QE1810 2Liu and Plagborg-Møller Quantitative Economics 14 (2023) So far, however, empirical work in this area has only been able to exploit limited features of the micro data sources that motivated the development of the new models. As emphasized by Ahn et al. (2017), the burgeoning academic literature has mostly calibrated model parameters and performed overidentification tests by matching a few empirical moments that are deemed important a priori. This approach may be highly inefficient, as it ignores that the models’ implied macro dynamics and cross-sectional properties often fully determine the entire distribution of the observed macro and micro data. The failure to exploit the joint information content of macro and micro data stands in stark contrast to the well-developed inference procedures for estimating representative agent models using only macro data (Herbst and Schorfheide (2016)). To exploit the full information content of macro and micro data, we develop a general technique to perform Bayesian inference in heterogeneous agent models. We assume the availability of aggregate time series data as well as repeated cross-sections of micro data. Evaluation of the joint macro and micro likelihood function is complicated by the fact that the model-implied cross-sectional distributions typically depend on unobserved aggregate state variables. To overcome this problem, we devise a way to compute a numerically unbiased estimate of the model-implied likelihood function of the macro and micro data. As argued by Andrieu, Doucet, and Holenstein (2010)andFlury and Shephard (2011), such an unbiased likelihood estimate can be employed in standard Markov Chain Monte Carlo (MCMC) procedures to generate draws from the fully efficient Bayesian posterior distribution given all available data. The starting point of our analysis is the insight that existing solution methods for heterogeneous agent models directly imply the functional form of the joint sampling distribution of macro and micro data, given structural parameters. These models are typically solved numerically by imposing a flexible functional form on the relevant cross-sectional distributions (e.g., a discrete histogram or parametric family of densities). The distributions are governed by time-varying unobserved state variables (e.g., moments). To calculate the model-implied likelihood, we decompose it into two parts. First, heterogeneous agent models are typically solved using the method of Reiter (2009), which linearizes with respect to the macro shocks but not the micro shocks. Hence, the macro part of the likelihood can be evaluated using standard linear state space methods, as proposed by Mongey and Williams (2017)andWinberry (2018).2Second, the likelihood of the repeated cross-sections of micro data, conditional on the macro state variables, can be evaluated by simply plugging into the assumed cross-sectional density. The key challenge that our method overcomes is that the econometrician typically does not directly observe the macro state variables. Instead, the observed macro time series are imperfectly informative about the underlying states. Our procedure can loosely be viewed as a rigorous Bayesian version of a twostep approach: First, we estimate the latent macro states from macro data, and then we compute the model-implied cross-sectional likelihood conditional on these estimated macro states. More precisely, we obtain a numerically unbiased estimate of the 2If non-Reiter model solution methods are used, our general estimation approach could in principle still be applied, though its computational feasibility would be context-dependent, as discussed in Section 7. Quantitative Economics 14 (2023) Estimation of heterogeneous agent models 3 likelihood by averaging the cross-sectional likelihood across repeated draws from the smoothing distribution of the hidden states given the macro data. We emphasize that, despite being based on a likelihood estimate, our method is fully Bayesian and automatically takes into account all sources of uncertainty about parameters and states. An attractive computational feature is that evaluation of the micro part of the likelihood lends itself naturally to parallel computing. Hence, computation time scales well with the size of the data set. Though our baseline method is designed for repeated cross-sections of micro data, we present ideas for exploiting panel data in Section 6. We perform finite-sample valid and fully efficient Bayesian inference by plugging the unbiased likelihood estimate into a standard MCMC algorithm. The generic arguments of Andrieu, Doucet, and Holenstein (2010)andFlury and Shephard (2011) imply that the ergodic distribution of the MCMC chain is the full-information posterior distribution that we would have obtained if we had known how to evaluate the exact likelihood function (not just an unbiased estimate of it). This is true no matter how many smoothing draws are used to compute the unbiased likelihood estimate. In principle, we may use any MCMC posterior sampling algorithm that relies only on evaluating (the unbiased estimate of) the posterior density, such as Random Walk Metropolis–Hastings. In contrast to other estimation methods, our full-information method is automatically finite-sample efficient and can easily handle unobserved individual heterogeneity, micro measurement error, as well as data imperfections such as selection or censoring. In an important early work, Mongey and Williams (2017) propose to exploit micro data by collapsing it to time series of cross-sectional moments and incorporating these into the macro likelihood. In principle, this approach can be as efficient as our fullinformation approach if the structural model implies that these moments are sufficient statistics for the micro data. We provide examples where this is not the case, for example, due to the presence of unobserved individual heterogeneity and/or micro measurement error. Even when sufficient statistics do exist, it is necessary to properly account for sampling error in the observed cross-sectional moments, which is done automatically by our full-information likelihood method, but could be delicate and imprecise for momentbased approaches. Moreover, textbook adjustments to the micro likelihood allow us to accommodate specific empirically realistic features of micro data such as selection (e.g., oversampling of large firms) or censoring (e.g., top-coding of income), whereas this is challenging to do efficiently with moment-based approaches. We illustrate the joint inferential power of macro and micro data through two numerical examples: a heterogeneous household model (Krusell and Smith (1998)) and a heterogeneous firm model (Khan and Thomas (2008)). In both cases, we assume that the econometrician observes certain standard macro time series as well as intermittent repeated cross-sections of, respectively, (i) household employment and income and (ii) firm capital and labor inputs. Using simulated data, and given flat priors, we show that our full-information method accurately recovers the true structural model parameters. Importantly, for several structural parameters, the micro data reduces the length of posterior credible intervals substantially, relative to inference that exploits only the macro data. In fact, we give examples of parameters that can only be identified if micro 4Liu and Plagborg-Møller Quantitative Economics 14 (2023) data is available. In contrast, inference from moment-based approaches can be highly inaccurate and sensitive to the choice of moments. We deliberately keep our numerical illustrations low-dimensional and build our code on top of the user-friendly Dynare-based model solution method of Winberry (2018). Though pedagogically useful, this particular numerical model solution method cannot handle very rich models, so a full-scale empirical illustration is outside the scope of this paper. However, there is nothing in our general inference approach that rules out larger-scale models. We argue in Section 7that our general inference approach is compatible with cutting-edge model solution methods that apply automatic dimension reduction of the state space equations (Ahn et al. (2017)). Literature Our paper contributes to the recent literature on structural estimation of heterogeneous agent models by exploiting the full, combined information content available in macro and micro data. We build on the idea of Mongey and Williams (2017)and Winberry (2018) to estimate heterogeneous agent models from the linear state space representation obtained from the Reiter (2009) model solution approach. Several papers have exploited only macro data (as well as calibrated steady-state micro moments) for estimation, including Winberry (2018), Hasumi and Iiboshi (2019), Auclert, Rognlie, and Straub (2020), Acharya, Chen, Del Negro, Dogra, Matlin, and Sarfati (2021), and Auclert, Bardóczy, Rognlie, and Straub (2021). Challe, Matheron, Ragot, and Rubio- Ramirez (2017), Mongey and Williams (2017), Bayer, Born, and Luetticke (2020), and Papp and Reiter (2020) additionally track particular cross-sectional moments over time. In contrast, we exploit the entire model-implied likelihood function given repeated micro cross-sections, which is (at least weakly) more efficient, as discussed further in Section 3.3. We are not aware of other papers that tackle the fundamental problem that the aggregate shocks affecting cross-sectional heterogeneity are not directly observed. Parra- Alvarez, Posch, and Wang (2020) use the model-implied steady-state micro likelihood in a heterogeneous household model, but abstract from macro data or aggregate dynamics. Closest to our approach are Fernández-Villaverde, Hurtado, and Nuño (2019), who exploit the model-implied joint sampling density of macro and micro data in a particular heterogeneous agent macro model. However, they assume that the underlying state variables are directly observed, whereas our contribution is to solve the computational challenges that arise in the generic case where the macro states are (partially) latent. Certain other existing methods for combining macro and micro data cannot be applied in our setting. Hahn, Kuersteiner, and Mazzocco (2022) develop asymptotic theory for estimation using interdependent micro and macro data sets, but their fullinformation approach requires derivatives of the exact likelihood in closed form, which is not available in our setting due to the need to integrate out unobserved state variables. Chang, Chen, and Schorfheide (2021)propose a reduced-form approach to estimating the feedback loop between aggregate time series and heterogeneous micro data; they do not consider estimation of structural models. In likelihood estimation of representative agent models, micro data has mainly been used to inform the prior, as in Chang, Gomes, and Schorfheide (2002). Finally, unlike the microeconometric literature on heterogeneous agent models (Arellano and Bonhomme (2017)), our work explicitly seeks to Quantitative Economics 14 (2023) Estimation of heterogeneous agent models 5 estimate the deep parameters of a general equilibrium macro model by also incorporating aggregate time series data. Outline Section 2shows that heterogeneous agent models imply a fully-specified statistical model for the macro and micro data. Section 3presents our method for computing an unbiased likelihood estimate that is used to perform efficient Bayesian inference. There we also compare our full-information approach with moment-based estimation approaches. Sections 4and 5illustrate the inferential power of combining macro and micro data using two simple numerical examples, a heterogeneous household model and a heterogeneous firm model. Section 6proposes an extension to panel data. Section 7concludes and discusses possible future research directions. Appendix A contains proofs and technical results. A Supplemental Appendix (Liu and Plagborg-Møller (2023)) and a full Matlab code suite are available online.3 2. Framework We first describe how heterogeneous agent models generically imply a statistical model for the macro and micro data. Then we illustrate how a simple model with heterogeneous households fits into this framework. 2.1 A general heterogeneous agent framework Consider a given structural model that implies a fully-specified equilibrium relationship among a set of aggregate and idiosyncratic variables. We assume the availability of macro time series data as well as repeated cross-sections of micro data, as summarized in Figure 1.Letx≡{xt}1≤t≤Tdenote the vector of observed time series data (e.g., real GDP growth), where xtis a vector, and Tdenotes the time series sample size. At a subset T⊂{1, 2, ,T}of time points, we additionally observe the micro data y≡{yi,t}1≤i≤Nt,t∈T,whereyi,tis a vector (e.g., the asset holdings of household ior the employment of firm i). At each time t, the cross-section {yi,t}1≤i≤Ntis sampled at random from the model-implied cross-sectional distribution conditional on some macro Figure 1. Diagram of the distribution of the macro and micro data implied by a heterogeneous agent model. The state vector ztincludes any time-varying parameters that govern the cross– sectional distribution p(yi,t|zt,θ). 3https://github.com/mikkelpm/het_agents_bayes 6Liu and Plagborg-Møller Quantitative Economics 14 (2023) state vector zt. For now, it is convenient to assume that {yi,t}constitutes a representative sample, but sample selection or censoring are easily accommodated in the framework, as we demonstrate in Section 5.4. Formally, we make the following assumption. Assumption 1. The data is sampled as follows: 1. Conditional on z≡{zt}T t=1,the micro data {yi,t}1≤i≤Nt,t∈Tis independent across t and the data points {yi,t}Nt i=1at time tare sampled i.i.d.from the density p(yi,t|zt,θ). 2. Conditional on z,the micro data yis independent of the macro data x. 3. Conditional on ztand {xτ,zτ}τ≤t−1,the macro data xtis sampled from the density p(xt|zt,θ).Conditional on {zτ}τ≤t−1,the state vector ztis sampled from the density p(zt|zt−1,θ). The first condition above operationalizes the notion of representative sampling of repeated cross-sections. The second condition entails no loss of generality, since we can always include xtin the state vector zt. The third condition is a standard Markovian state space formulation of the aggregate dynamics, as discussed further below. Given the structural parameter vector θ, the fully-specified heterogeneous agent model implies functional forms for the macro observation density p(xt|zt,θ),the macro state transition density p(zt|zt−1,θ), and the micro sampling density p(yi,t| zt,θ). These density functions reflect the equilibrium of the model, as we illustrate in the next subsection, and they are the key inputs in the likelihood computation in Section 3. Notice that the framework allows the micro and macro data to be dependent, though this dependence must be fully captured by the macro state vector zt,which is determined by the structure of the model at hand. Because the sampling densities p(xt|zt,θ)and p(yi,t|zt,θ)are derived from an equilibrium model, the likelihood function derived below in equation (5) automatically embodies any constraints of the type envisioned by Imbens and Lancaster (1994) on the relationship between the aggregate macro data and the time-varying population moments of the micro sampling distribution. For example, if yi,tequals individual-level consumption, xtequals aggregate consumption, and ztequals the underlying macro shocks (which determine the dynamics of aggregates and of the micro distribution), then the asymptotic adding-up constraint that limNt→∞ 1 NtNt i=1yi,t a.s. =xtwill be automatically satisfied if the sampling densities are derived from a model that imposes market clearing. In most applications, some of the aggregate state variables ztthat influence the macro and micro sampling densities are unobserved, that is, zt= xt.Thisfactcomplicates the evaluation of the exact likelihood function and is the key technical challenge that we overcome in this paper, as discussed in Section 3. 2.2 Example: Heterogeneous household model We use a simple heterogeneous household model à la Krusell and Smith (1998)toillustrate the components of the general framework introduced in Section 2.1.Ourdiscussion of the model and the numerical equilibrium solution technique largely follows Quantitative Economics 14 (2023) Estimation of heterogeneous agent models 7 Winberry (2016,2018). Though this model is far too stylized for quantitative empirical work, we demonstrate the flexibility of our framework by adding complications such as permanent heterogeneity among households as well as measurement error in observables. In Section 4, we will estimate a calibrated version of this model on simulated data. Model assumptions A continuum of heterogeneous households i∈[0, 1]are exposed to idiosyncratic employment risk as well as aggregate shocks to wages and asset returns. Households have log preferences over consumption ci,tat time t=0, 1, 2, .Whenemployed (i,t=1), households receive wage income net of an income tax levied at rate τ. When unemployed (i,t=0), they receive unemployment benefits equal to a fraction b of their hypothetical working wage. The idiosyncratic unemployment state i,tevolves exogenously according to a two-state first-order Markov process that is independent of aggregate conditions and household decisions. Households cannot insure themselves against their employment risk, since the only available financial instruments are shares of capital ˜ ai,t, which yield a rate of return rt. Financial investment is subject to the borrowing constraint ˜ ai,t≥0. For expositional purposes, we add a dimension of permanent household heterogeneity: Each household is endowed with a permanent labor productivity level λi,which is drawn at the beginning of time from a log-normal distribution with mean parameter E[logλi]=μλ≤0 and variance parameter chosen such that E[λi]=1. An employed household inelastically supplies λiefficiency units of labor, earning pretax income of λiwt,wherewtis the real wage per efficiency unit of labor. To summarize, the households’ problem can be written max ci,t,ai,t≥0 E0∞  t=0 βtlogci,t s.t. ci,t=λiwt(1−τ)i,t+b(1−i,t)+(1+rt)ai,t−1−ai,t, where ai,t=˜ ai,t/λiare the normalized asset holdings. A representative firm produces the consumption good using a Cobb–Douglas production function Yt=eζtKα tL1−α, where aggregate capital Ktdepreciates at rate δ,and Lis the aggregate level of labor efficiency units (which is constant over time since employment risk is purely idiosyncratic). The firm hires labor and rents capital in competitive input markets. Log total factor productivity (TFP) evolves as an AR(1) process ζt=ρζζt−1+εt,whereεt i.i.d. ∼N(0, σ2 ζ). The government balances its budget period by period, implying τL =b(1−L). We collect the deep parameters of this model in the vector θ. These include β,α,δ, τ,ρζ,σζ, the transition probabilities for idiosyncratic employment states, and μλ. Equilibrium definition and computation The mathematical definition of a recursive competitive equilibrium is standard, and we refer to Winberry (2016) for details. We now review Winberry’s method for solving the model numerically. A key model object is the cross-sectional joint distribution of the micro state variables, that is, employment status i,t, normalized assets ai,t−1, and permanent produc- 8Liu and Plagborg-Møller Quantitative Economics 14 (2023) tivity λi. This distribution, which we denote ˜μt(,a,λ), is time-varying as it implicitly depends on the aggregate productivity state variable ζtat time t. Due to log utility and the linearity of the households’ budget constraint in λi, macro aggregates are unaffected by the distribution of the permanent cross-sectional heterogeneity λi(recall that E[λi]=1). This implies that the mean parameter μλof the log-normal distribution of λiis only identifiable if micro data is available, as discussed further in Section 4. In equilibrium, we have ˜μt(,a,λ)=μt(,a)F(λ|μλ),whereF(·|μλ)denotes the time-invariant lognormal distribution for λi. To solve the model numerically, Winberry (2016,2018) assumes that the infinitedimensional cross-sectional distribution μt(,a)can be well approximated by a rich but finite-dimensional family of distributions. The distribution of agiven is a mixture of a mass point at 0 (the borrowing constraint) and an absolutely continuous distribution concentrated on (0, ∞). At every point in time, Winberry approximates the absolutely continuous part using a density of the exponential form g(a)=exp˜ϕ0+˜ϕ1˜ m1+ q  =2 ϕ(a−˜ m1)−˜ m, where ˜ m1=E[a|],˜ ml =E[(a−˜ m1)|]for l≥2, the ˜ϕ’s are coefficients of the distribution, and q∈Nis a tuning parameter that determines the quality of the numerical approximation. The q+1 coefficients {˜ϕ}0≤l≤qare pinned down by the qmoments {˜ m}1≤l≤q, along with the normalization that g(a)integrates to one. The approximation of the distribution μt(,a)at any point in time therefore depends on 2(q+1)parameters: the probability point mass at a=0aswellastheqmoments {˜ m}1≤l≤q,for each employment state . Denote the vector of all these parameters by ψ.Themodelsolution method proceeds under the assumption μt(a,)=G(a,;ψt),whereGdenotes the previously specified parametric mixture functional form for the distribution, and we have added a time subscript to the parameter vector ψ=ψt. Though the approximation μt(a,)≈G(a,;ψt)only becomes exact in the limit q→∞, the approximation may be good enough for small qto satisfy the model’s equilibrium equations to a high degree of numerical accuracy. Adopting the distributional approximation, the model’s aggregate equilibrium can now be written as a nonlinear system of expectational equations in a finite-dimensional vector ztof macro variables: EtH(zt+1,zt,εt+1;θ)=0, (1) where we have made explicit the dependence on the deep model parameters θ.Consistent with the notation in Section 2.1, the vector ztincludes (log) aggregate output log(Yt), capital log(Kt), wages log(wt), rate of return rt, and productivity ζt,butalsothe time-varying distributional parameters ψt. For brevity, we do not specify the full equilibrium correspondence H(·)here but refer to Winberry (2016) for details. Among other things, H(·)enforces that the evolution over time of the cross-sectional distributional parameters ψtis consistent with households’ optimal savings decision rule, given the Quantitative Economics 14 (2023) Estimation of heterogeneous agent models 15 4. Illustration:Heterogeneous household model We now demonstrate that combining macro and micro data can sharpen structural inference when estimating the heterogeneous household model of Section 2.2 on simulated data. We contrast the results of our efficient full-information approach with those of an alternative moment-based approach. This section should be viewed as a proof-of- concept exercise, as we deliberately keep the dimensionality of the inference problem small in order to focus attention on the core workings of our procedure. 4.1 Model, data, and prior We consider the stylized heterogeneous household model defined in Section 2.2.Weaim to estimate the households’ discount factor β, the standard deviation σeof the measurement error in log output, and the individual productivity heterogeneity parameter μλ. All other parameters are assumed known for simplicity. Consistent with Section 2.2, we assume that the econometrician observes aggregate data on log output with measurement error, as well as repeated cross-sections of household employment status i,tand after-tax/after-benefits income ιi,t. We adopt the annual parameter calibration in Winberry (2016); see Supplemental Appendix C.1. In particular, β=0.96. We choose the true measurement error standard deviation σeso that about 20% of the variance of observed log output is due to measurement error, yielding σe=0.02.7The individual heterogeneity parameter μλis chosen to be −0.25, implying that the model’s cross-sectional 20th to 90th percentile range of log after-tax income roughly matches the range in U.S. data (Piketty, Saez, and Zucman (2018, Table I)). Using this calibration, we simulate T=100periodsofmacrodata,aswellasmicro data consisting of Nt=N=1000 households observed at each of the ten time points t= 10, 20, 30, , 100. The data is simulated using the same approximate model solution method as is used to compute the unbiased likelihood estimate; see Section 2.2. Finally, we choose the prior on (β,σe,μλ)to be flat in the natural parameter space. 4.2 Computation Following Winberry (2016,2018), we solve the model using a Dynare implementation of the Reiter (2009) method. This allows us to use Dynare’s built-in Kalman filter/smoother procedures when evaluating the micro likelihood estimate (6). We use an approximation of degree q=3 when approximating the asset distribution, in the notation of Section 2.2. We average the likelihood across J=500 smoothing draws. The integral (4)inthemicro sampling density of income is evaluated using a combination of numerical integration and interpolation.8To simulate micro data from the cross-sectional distribution, we 7One possible real-world interpretation of the measurement error is that it represents the statistical uncertainty in estimating the natural rate of output (recall that the model abstracts from nominal rigidities). 8First, we use a univariate numerical integration routine to evaluate the integral on an equal-spaced grid of values for log ι. Then we use cubic spline interpolation to evaluate the integral at arbitrary ι.Inpractice, a small number of grid points is sufficient in this application, since the density (4) is a smooth function of ι. 16 Liu and Plagborg-Møller Quantitative Economics 14 (2023) apply the inverse probability transform to the model-implied cumulative distribution function of assets, which in turn is computed using numerical integration. For simplicity, our MCMC algorithm is a basic Random Walk Metropolis-Hastings algorithm with tuned proposal covariance matrix and adaptive step size (Atchadé and Rosenthal (2005)).9The starting values are determined by a rough grid search on the simulated data. We generate 10,000 draws and discard the first 1000 as burn-in. Using parallel computing on 20 cores, likelihood evaluation takes about as long as Winberry’s (2016) procedure for computing the model’s steady state. 4.3 Results Figure 2shows that both macro and micro data can be useful or even essential for estimating some parameters, but not others. The figure depicts the posterior densities of the three parameters, on a single sample of simulated data. The full-information posterior (solid curves) is concentrated close to the true values of the three parameters (which are marked by vertical thin dashed lines). The figure also shows the posterior density without conditioning on the micro data (dashed curves). The household discount factor β is an important determinant of not just aggregate variables, but also the heterogeneous actions of the micro agents in the economy. Ignoring the micro data leads to substantially less accurate inference about βin this simulation, as the macro-only posterior is less precisely centered around the true value as well as more diffuse than the fullinformation posterior. Nevertheless, macro data clearly does meaningfully contribute to Figure 2. Heterogeneous household model: Posterior density. Posterior densities with (solid curves) and without (dashed curves) conditioning on the micro data. Both sets of results use the same simulated data set. Vertical dashed lines indicate true parameter values. Posterior density estimates from the 9000 retained MCMC draws using Matlab’s ksdensity function with default settings. The third display omits the macro-only results, since μλis not identified from macro data alone. 9Our proposal distribution is a mixture of (i) the adapted multivariate normal distribution and (ii) a diffuse normal distribution, with 95% probability attached to the former. We verified the diminishing adaption condition and containment condition in Rosenthal (2011), so the distribution of the MCMC draws will converge to the posterior distribution of the parameters. Quantitative Economics 14 (2023) Estimation of heterogeneous agent models 17 Figure 3. Heterogeneous household model: Consumption policy function, employed. Estimated steady-state consumption policy function for employed households, either using both micro and macro data (left panel) or only using macro data (right panel). The thick curve is computed under the true parameters. The thin lines are 900 posterior draws (computed using every 10th MCMC draw after burn-in). X-axes are normalized asset holdings ai,t. pinning down the parameter β.Morestarkly,μλcan only be identified from the crosssection, since by construction the macro aggregates are not influenced by the distribution of the individual permanent productivity draws λi. In contrast, essentially all the information about the measurement error standard deviation σecomes from the macro data, again by construction. Thus, our results here illustrate the general lesson that both macro and micro data can be either essential, useful, or irrelevant for estimating different parameters. Figure 3shows that efficient use of the micro data leads to substantially more precise estimates of the steady-state consumption policy function for employed households.10 The left panel shows that full-information posterior draws of the consumption policy function (thin curves) are fairly well centered around the true function (thick curve), as is expected given the accurate inference about βdepicted in Figure 2. In contrast, the right panel shows that macro-only posterior draws are less well centered and exhibit higher variance, especially for households with high or low current asset holdings. The added precision afforded by efficient use of the micro data translates into more precise estimates of the marginal propensity to consume (the derivative of the consumption policy function) at the extremes of the asset distribution. This is potentially useful when analyzing the two-way feedback effect between macroeconomic policies and redistribution (Auclert (2019)). The extra precision afforded by micro data also sharpens inference on the impulse response function of the asset distribution with respect to an aggregate productivity shock. Figure 4shows full-information (left panel) and macro-only (right panel) posterior draws of the impulse response function of employed households’ asset holding density, in the periods following a 5% aggregate productivity shock.11 Once again, the 10Figure C.1 in Supplemental Appendix C.2 plots the policy function for unemployed households. 11For unemployed households, see Figure C.2 in Supplemental Appendix C.2. 18 Liu and Plagborg-Møller Quantitative Economics 14 (2023) Figure 4. Het. household model: Impulse responses of asset distribution, employed. Estimated impulse response function of employed households’ asset distribution with respect to an aggregate productivity shock, either using both micro and macro data (left panel) or only using macro data (right panel). The thin lines are 900 posterior draws (computed using every 10th MCMC draw after burn-in). X-axes are normalized asset holdings ai,t. The four rows in each panel are the asset densities at impulse response horizons 0 (impact), 2, 4, and 8. The dashed and thick solid curves are the steady-state density and the impulse response, respectively, computed under the true parameters. On impact the true impulse response equals the steady-state density, since households’ portfolio choice is predetermined. full-information results have substantially lower variance. Following the shock, there is a noticeable movement of the asset distribution computed under the true parameters (thick solid curve). At horizon h=8, the mean increases by 0.16 relative to the steady state (dashed curve), the variance increases by 0.10, and the third central moment decreases by 0.06. However, the true movement in the asset distribution is not so large relative to the estimation uncertainty. This further motivates the use of an efficient inference method that validly takes into account all estimation uncertainty. The previous qualitative conclusions hold up in repeated simulations from the calibrated model. We repeat the MCMC estimation exercise on 10 different simulated data sets.12 Figure 5plots all 10 full-information and macro-only posterior densities for the three parameters on the same plot. Notice that the full-information densities for βsystematically concentrate closer to the true value than the macro-only posteriors do, as in Figure 2. Our inference approach is valid in the usual Bayesian sense no matter how small the sample size is. In Figure C.3 of Supplemental Appendix C.2, we show that the fullinformation approach still yields useful inference about the model parameters if we only observe N=100 observations every ten periods (instead of N=1000 as above). 4.4 Comparison with moment-based methods In this subsection, we compare the above full-information results with a moment-based inference approach, to shed light on the theoretical comparison in Section 3.3.Due 12Computational constraints preclude a full Monte Carlo study. Quantitative Economics 14 (2023) Estimation of heterogeneous agent models 19 Figure 5. Heterogeneous household model: Posterior density, multiple simulations. Posterior densities with (light curves) and without (dark curves) conditioning on the micro data, for 10 different simulated data sets. See also caption for Figure 2. to the unobserved individual heterogeneity parameter λi, fixed-dimensional sufficient statistics do not exist in this model with the given observables.13 Hence, we follow empirical practice and compute an ad hoc selection of cross-sectional moments, including the sample mean, variance, and third central moment of household after-tax income. We compute the moments separately for the groups of employed and unemployed households, in each period t=10, 20, , 100 where micro data is observed. We consider three moment-based approaches with different numbers of observables: The “1st Moment” approach only incorporates time series of sample means, the “2nd Moment” approach incorporates both sample means and variances, and the “3rd Moment” approach incorporates sample moments up to the third order. Once we compute the time series of cross-sectional moments on the simulated data, we treat them as additional time series observables and proceed as in the “Macro Only” approach considered earlier. To account for the sampling uncertainty in the crosssectional moments, we appeal to a central limit theorem and treat the moments as jointly Gaussian, which is equivalent to adding measurement error in those state space equations that correspond to the moments. A natural and practical way to construct the variance-covariance matrix of the measurement error is to estimate its elements using higher-order sample moments of micro data. The variance-covariance matrix is actually time-varying according to the structural model, but since this would be challenging to account for, we treat it as fixed over the sample.14 Supplemental Appendix B provides the details of how we estimate the variance-covariance matrix. The computation time of the moment-based likelihood functions is not much faster than our full-information approach, since the evaluation of the micro likelihood (which is specific to the fullinformation method) takes approximately the same amount of time as the calculation 13The unobserved individual heterogeneity is observationally equivalent to micro measurement error given repeated cross-sections of micro data. 14Higher-order sample moments are less accurate approximations to their population counterparts. Given an empirically relevant cross-sectional sample size, the resulting variance-covariance matrix would be even more imprecise if inferred period by period. 20 Liu and Plagborg-Møller Quantitative Economics 14 (2023) Figure 6. Heterogeneous household model: Likelihood comparison. Comparison of log likelihoods across inference methods, based on one typical simulated data set. Each panel depicts univariate deviations of a single parameter while keeping all other parameters at their true values. The maximum of each likelihood curve is normalized to be zero. Vertical dashed lines indicate true parameter values. The “1st Moment” and “Macro Only” curves are flat on the right panel, since μλis not identified from this data alone. For results across 10 different simulated data sets, see Figure A.1 in Appendix A.4. of the model’s steady state (which is common to all methods), when implemented on a research cluster with 20 parallel workers. We compare the shape and location of the likelihood functions for the full-informa- tion and moment-based methods.15 For graphical clarity, we vary a single parameter at a time, keeping the other parameters at their true values. Figure 6plots the univariate log likelihood functions of all inference approaches based on one typical simulated data set.16 Since we are interested in the curvature of the likelihood functions near their maxima, and not the overall level of the functions, we normalize each curve by subtracting its maximum. Figure 6shows that the moment-based likelihoods do not approximate the efficient full-information likelihood well, with the “3rd Moment” likelihood being particularly inaccurately centered. There are two reasons for this. First, as discussed in Section 3.3, there is no theoretical sufficient statistics in this setup, so all the moment-based approaches incur some efficiency loss. Second, the sampling distributions of higher-order sample moments are not well approximated by Gaussian distributions in finite samples, and the measurement error variance-covariance matrix depends on even higher-order moments, which are poorly estimated. A separate issue is that the individual heterogeneity parameter μλcannot even be identified using the “1st Moment” approach, since this parameter does not influence first moments of the micro data. The “2nd Moment” 15We omit full posterior inference results for the moment-based methods, as they were more prone to MCMC convergence issues than our full-information method. 16We compute the full-information likelihood function by averaging across J=500 smoothing draws. For a clearer comparison of the plotted likelihood functions, we fix the random numbers used to draw from the smoothing distribution across parameter values. Note that we do not fix these random numbers in the MCMC algorithm, as required by the Andrieu, Doucet, and Holenstein (2010)argument. Quantitative Economics 14 (2023) Estimation of heterogeneous agent models 21 likelihood is not entirely misleading but nevertheless differs meaningfully from the fullinformation likelihood.17 Figure A.1 in Appendix A.4 confirms that the aforementioned qualitative conclusions hold up across 10 different simulated data sets. To summarize, even in this relatively simple model, the moment-based methods we consider lead to a poor approximation of the full-information likelihood, and the inference can be highly sensitive to the choice of which moments to include. It is possible that other implementations of the moment-based approach would work better in particular applications. Nevertheless, any moment-based approach will require challenging ad hoc choices, such as which moments to use and how to account for their sampling uncertainty. No such choices are required by the efficient full-information approach developed in this paper. 5. Illustration:Heterogeneous firm model As our second proof-of-concept example, we estimate a version of the heterogeneous firm model of Khan and Thomas (2008). In addition to showing that our general inference approach can be applied outside the specific Krusell and Smith (1998) family of models, we use this section to illustrate how sample selection or data censoring can easily be accommodated in our method. 5.1 Model, data, and prior A continuum of heterogeneous firms are subject to both idiosyncratic and aggregate productivity shocks. Investment is subject to nonconvex adjustment costs. Specifically, firm i’s investment Ii,tis free if |Ii,t/ki,t|≤a,whereki,tis the firm-specific capital stock, and a≥0 is a parameter. Otherwise, firms pay a fixed adjustment cost of ξi,tin units of labor. ξi,tis drawn at the beginning of every period from a uniform distribution on the interval [0, ¯ ξ], independently across firms and time. Here, ¯ ξ≥0 is another parameter. In addition to the aggregate productivity shock, there is a second aggregate shock that affects investment efficiency. The representative household has additively separable preferences over log consumption and (close to linear) leisure time. For brevity, we relegate the details of the model to Supplemental Appendix D.1, which entirely follows Winberry’s (2018) version of the Khan and Thomas (2008)model. We aim to estimate the adjustment cost parameters ¯ ξand a.Khan and Thomas (2008) showed that these parameters have little impact on the aggregate macro implications of the model in their preferred calibration; hence, micro data is needed. We keep all other parameters fixed at their true values for simplicity. Supplemental Appendix D.3 provides results for an alternative exercise where we instead estimate the parameters of the firms’ idiosyncratic productivity process; the key messages are qualitatively similar to those presented below. 17The “Full Info” and “Macro Only” likelihoods are consistent with the posterior densities plotted in Section 4.3.Forβ, the “Macro Only” likelihood has a smaller curvature around the peak and a wider range of peaks across simulated data sets, so the full information method helps sharpen the inference of β.For σe, the “Macro Only” curves are close to their “Full Info” counterparts. The parameter μλis not identified in the “Macro Only” case, so the corresponding likelihood function is flat. 22 Liu and Plagborg-Møller Quantitative Economics 14 (2023) We adopt the annual calibration of Winberry (2018),whichinturnfollowsKhan and Thomas (2008); see Supplemental Appendix D.2. However, we make an exception in setting the firm’s idiosyncratic log productivity AR(1) parameter ρ=0.53, following footnote 5 in Khan and Thomas (2008).18 We then adjust the log productivity innovation standard deviation σ=0.0364, so that the variance of the idiosyncratic log productivity process is unchanged from the baseline calibration in Khan and Thomas (2008)and Winberry (2018). The macro implications of our calibration are virtually identical to the baseline in Khan and Thomas (2008), as those authors note. We assume that the econometrician observes time series on aggregate output and investment, as well as repeated cross-sections of micro data on firms’ capital and labor inputs. We simulate macro data with sample size T=50, while micro cross-sections of size N=1000 are observed at each point in time t=1, , 50. Unlike in Section 4,wedo not add measurement error to the macro observables. The prior on (¯ ξ,a)is chosen to be flat in the natural parameter space. 5.2 Computation As in Section 4, we solve and simulate the model using the Winberry (2018)Dynare solution method. We follow Winberry (2018) and approximate the cross-sectional density of the firms’ micro state variables (log capital and idiosyncratic productivity) with a multivariate normal distribution. Computation of the micro sampling density is simple, since—conditional on macro states—the micro observables (capital and labor) are loglinear transformations of these micro state variables. We use J=500 smoothing draws to compute the unbiased likelihood estimate. The MCMC routine is the same as in Section 4. The starting values are selected by a rough grid search on the simulated data. We generate 10,000 draws and discard the first 1000 as burn-in. Likelihood evaluation using 20 parallel cores is several times faster than computing the model’s steady state. 5.3 Results Despite the finding in Khan and Thomas (2008) that macro data is essentially uninformative about the firms’ adjustment cost parameters, these are accurately estimated when the micro data is used also. Figure 7shows the posterior densities of ¯ ξand acomputed on 10 different simulated data sets. The posterior distribution of each parameter is systematically concentrated close to the true parameter values. We refrain from visually comparing these results with inference that relies only on macro data, since the macro likelihood is almost entirely flat as a function of (¯ ξ,a), consistent with Khan and Thomas (2008).19 Thus, micro data is essential to inference about these parameters. This finding is broadly consistent with Bachmann and Bayer (2014), who show that the dynamics of the cross-sectional dispersion of firm investment are very informative about the nature of firm-level frictions. 18This avoids numerical issues that arise when solving the model for high degrees of persistence, as required in the estimation exercise in Supplemental Appendix D.3. 19On average across the 10 simulated data sets, the standard deviation (after burn-in) of the macro log likelihood logp(x|θ)across all Metropolis–Hastings proposals of the parameters is only 0.14, while it is 18.7 for the micro log likelihood log p(y|x,θ). Quantitative Economics 14 (2023) Estimation of heterogeneous agent models 23 Figure 7. Heterogeneous firm model: Posterior density, multiple simulations. Posterior densities across 10 simulated data sets. Vertical dashed lines indicate true parameter values. Posterior density estimates from the 9000 retained MCMC draws using Matlab’s ksdensity function with default settings. 5.4 Correcting for imperfect sampling of micro data One advantage of the likelihood approach adopted in this paper is that standard techniques can be applied to correct for sample selection or censoring in the micro data. This is highly relevant for applied work, since household or firm surveys are often subject to known data imperfections, even beyond measurement error. Valid inference about structural parameters merely requires that the micro sampling density p(yi,t|zt,θ)introduced in Section 2.1 accurately reflects the sampling mechanism, including the effects of selection or censoring. Hence, if it is known, say, that an observed variable such as household income is top-coded (i.e., censored) at the threshold ¯ y, then the functional form of the density p(yi,t|zt,θ)should take into account that the observed data equals a transformation yi,t=min{˜ yi,t,¯ y}of the theoretical household income ˜ yi,tin the DSGE model. The likelihood functions of such limited dependent variable sampling models are well known and readily looked up; see, for example, Wooldridge (2010, Chapters 17 and 19).20 We provide one illustration below. Other approaches to estimating heterogeneous agent models do not handle data imperfections as easily or efficiently. For example, inference based on cross-sectional moments of micro observables may require lengthy derivations to adjust the moment formulas for selection or censoring, especially for higher moments. Moreover, even in models where low-dimensional sufficient statistics exist for the underlying micro variables (cf. Section 3.3), the imperfectly observed micro data may not afford such sufficient 20If the nature of the data imperfection is only partially known, it may be possible to estimate the sampling mechanism from the data. For example, if the data is suspected to be subject to endogenous sample selection, one could specify a Heckman-type selection model and estimate the parameters of the selection model as part of the likelihood framework (Wooldridge (2010, Chapter 19)). It is outside the scope of this paper to consider nonparametric approaches or to analyze the consequences of misspecification of the sampling mechanism. 24 Liu and Plagborg-Møller Quantitative Economics 14 (2023) statistics. In contrast, our likelihood-based approach is automatically efficient, and the adjustments needed to account for common types of data imperfections can be looked up in microeconometrics textbooks. Illustration: Selection on outcomes We illustrate the previous points by adding an endogenous selection mechanism to the sampled micro data in the heterogeneous firm model. Assume that instead of observing a representative sample of firms every period, we observe the draws for those firms whose employment in that period exceeds the 90th percentile of the steady-state cross-sectional distribution of employment. To make the effective micro sample size comparable to that in Section 5.3, we here set the per-period micro sample size before selection equal to N=10,000. That is, out of 10,000 potential draws in a period, we only observe the capital and labor inputs of the approximately 1000 largest firms. Though stylized, this sampling mechanism is intended to mimic the real-world phenomenon that databases such as Compustat tend to only cover the largest active firms in the economy. To adjust the likelihood for selection, we combine the model-implied cross-sectional distribution of the idiosyncratic state variables with the functional form of the selection mechanism. Let gt(,k)be the cross-sectional distribution of idiosyncratic log productivity i,tand log capital ki,tat time t, implied by the model (this density is approximated using an exponential family of densities, as in Winberry (2018)). In the model, log employment is given by ni,t=(logν+ζt−log(wt)+i,t+αki,t)/(1−ν),wherewtis the aggregate wage, ζtis log aggregate TFP, and νand αare the output elasticities of labor and capital in the firm production function (ν+α<1). Since observations yi,t=(ni,t,ki,t) are observed if and only if ni,t≥¯ n, the micro sampling density is given by the truncation formula21 p(ni,t,ki,t|zt,θ)=(1−ν)gt(1−ν)ni,t−αki,t−logν−ζt+log(wt),ki,t ∞ −∞ ∞ −∞ 1logν+ζt−log(wt)++αk ≥(1−ν)¯ ngt(,k)ddk . The selection threshold ¯ nis given by the true 90th percentile of the steady-state distribution of log employment. We assume this threshold is known to the econometrician for simplicity.22 Figure 8shows the posterior distribution of the adjustment cost parameters (¯ ξ,a) in the model with selection, across 10 simulated data sets. All settings are the same as in Section 5.3, except for (i) the selection mechanism in the simulated micro data and the requisite adjustment to the functional form of the micro likelihood function, and (ii) the pre-selection micro sample size N=10,000 (as discussed above). The posterior distributions of the parameters of interest remain centered close to the true parameter values, with no appreciable increase in posterior uncertainty relative to Figure 7.This example demonstrates that data imperfections can be handled in a valid and efficient manner using standard likelihood techniques. 21The integral in the denominator can be computed in closed form if the density gt(,k)is multivariate Gaussian, which is the approximation we use in our numerical experiments, following Winberry (2018). 22In principle, ¯ ncould be treated as another parameter to be estimated from the available data. Quantitative Economics 14 (2023) Estimation of heterogeneous agent models 31 sufficient statistics do exist. For example, if si,t=B1(zt,θ)ϒ(yi,t)+B0(zt,θ)+B2(zt,θ)λi and both p(si,t|mt)and p(λi|θ)follow Gaussian distributions. (iii) ds>d y: For example, suppose si,tis two-dimensional whereas yi,tis onedimensional, say yi,t=s1,i,t,yi,t=s1,i,t+s2,i,t,oryi,t=s1,i,ts2,i,t.Wecanfirst expand the yi,tin (9)to ˜ yi,t=(yi,t,s2,i,t)and then integrate out s2,i,t. However, after the integration, the resulting micro likelihood as a function of yi,tmay not take the exponential family form anymore. A.3 Sampling distribution of cross-sectional moments: Example As alluded to in Section 3.3, here is a simple example demonstrating that p(ˆ mt|Nt,zt,θ) is not linear Gaussian in finite samples and, therefore, neither is p(ˆ m|N,x,θ). Suppose yi,t=si,tis a scalar and p(si,t|mt)is Gaussian, that is, a second-order exponential polynomial. Let ˆ m1,t=1 Nt Nt  i=1 si,tand ˆ m2,t=1 Nt Nt  i=1 (si,t−ˆ m1,t)2, with m1,tand m2,tbeing their population counterparts. Then standard calculations yield p(ˆ mt|Nt,zt,θ)=p(ˆ mt|mt)=φˆ m1,t;m1,t,m2,t Ntpχ2Ntˆ m2,t m2,t ;Nt−1, where φ(x;μ,σ2)represents the probability distribution function (pdf) of a Gaussian distribution with mean μand variance σ2,andpχ2(x;ν)isthepdfofachi-squareddis- tribution with νdegrees of freedom. We can see that the latter is not linear Gaussian. Moreover, when p(si,t|mt)follows a higher order exponential polynomial, the characterization of p(ˆ mt|Nt,zt,θ)would be even more complicated without a closed-form expression. A.4 Heterogeneous household model: Likelihood comparison Complementing the results for a single simulated data set in Section 4.4, Figure A.1 compares log likelihoods for the different inference methods across 10 different simulated data sets. Here, different inference methods are exhibited in different rows. Similar to Figure 6, each column depicts univariate deviations of a single parameter while keeping all other parameters at their true values. There are 10 likelihood curves in each panel, corresponding to the 10 simulated data sets. The maximum of each likelihood curve is normalized to be zero. Vertical dashed lines indicate true parameter values. The “1st Moment” and “Macro Only” curves are flat on the right panels of the second and the last rows, since μλis not identified from this data alone. We conclude from the figure that the full-information likelihood is systematically well centered and tightly concentrated around the true parameter values, whereas the various moment-based likelihoods are poorly centered, exhibit less curvature, and/or shift around substantially across simulations. 32 Liu and Plagborg-Møller Quantitative Economics 14 (2023) Figure A.1. Het. household model: Likelihood comparison, multiple simulations. Comparison of log likelihood functions across 10 different simulated data sets. See the description in Appendix A.4. 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